Neutrons / Second

Our last fusion run produced results we can use to calculate the neutrons per second emanating from the fusor.

To help with converting from bubbles to neutrons per second I contacted Rob Noulty of Bubble Technology Industries – makers of the bubble dosemeter we are using. He says:

Hi Mark:

Did you run a control detector (to see what natural background bubbles you will get over this time)? You must subtract these bubbles out assuming you have two detectors of roughly the same response (or you will need to scale).

Please note as well that you have a very small number of bubbles resulting in poor statistics (and a very large error). Based on 4 bubbles, the expected error is roughly 50%.

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3 responses

27062010

ingo(06:42:31) :

If your result is based on the 4 bubbles for which he states an error of 50% applies, you’ll have 728.4 +/- 364.2 fusion events / second (min 364.2, max 992.6) – I assume … but how does one derive the 50% on 4 bubbles ?
Ah, I see, it’s from the standard deviation: just 1/sqrt(N) (sigma_mean = sigma/sqrt(N), I assume sigma == 1 here)

your background subtraction is going to be way inside the error of the number of bubbles you detected. It looks all the operations are linear multiplictaions, so your plus/minus is exactly what ingo says.

Also, you may want to consider the factor of what your sensor is measuring – presumably you put it 7 cm away from the reactor, but how did you orient your sensor, (radially, tangentially) etc. Although again, this stuff will all fall in the error of your number of bubbles detected.

Your calculation looks good. I’m glad you got help from Rob Noulty; he is certainly a good source of information.

Statistical uncertainty is what Rob means when he says you have a 50% error (or equivalently, the example I made in a post upthread of 400 +/- 200 n/s). There are two ways to derive the uncertainty–experimentally, by making many trials of the same experiment and calculating the variance, which is impractical under the circumstances; or by application of the appropriate statistical model to calculate the “standard deviation”. Because the bubble detector is binary–when a neutron interacts you either get a count (bubble) or you get no counts–the appropriate statistical model is the binomial distribution, whose standard deviation is

sigma = SQRT(x * [1 – p])

where x is the experimentally-measured value, e.g. 4 bubs, and p is the probability of detecting a neutron incident on the BTI. Upon the further simplification that p is small, which it is since on the order of one in a million neutrons incident will form a bubble, the standard deviation simplifies to

sigma = SQRT(x) = SQRT(4 bubs) = 2 bubs

This practically means if you were to repeat this same experiment many times, you would expect various results (as calculated from the Poisson CDF, see Wikipedia’s Poisson distribution article):

0 bubs: once per 56 experiments
1 bubs: once per 14 experiments
2 bubs: once per 7 experiments
3 bubs: once per 5 experiments
4 bubs: once per 5 experiments
5 bubs: once per 6 experiments
6 bubs: once per 10 experiments
[etc.]

Quantities that are derived from the measurement, e.g. neutron source rate, total fusion rate, carry uncertainty propagated through the calculation from all the constituent experimental values. If there is a lot of uncertainty in the detector’s effective position (and there will be, because it is close to the source and has considerable geometric extent), that will enter the calculation too. Calculating the total number of fusions by multiplying the neutron source rate by two (to account for the aneutronic H-2(d,p) reaction) is only a very coarse approximation.

Convention is to report values rounded to the nearest statistically-significant figure. If the neutron source rate is 364.2 n/s and the uncertainty is 182 n/s, the reported value is “400 +/- 200 n / s.”